Tuning as Optimisation: A Quantitative Analysis of Equal and Non-Equal Temperaments

Twelve-tone equal temperament (12-TET) is often described as a “compromise”tuning that enables unrestricted modulation at the cost of harmonic impurity. Thisclaim is usually qualitative: it is rarely clear what is being optimised, which intervalsare prioritised, or what loss function is being assumed. This paper develops a frame-work for making such comparisons precise. Using logarithmic pitch representation incents, I define interval-specific approximation errors relative to just-intonation tar-gets and aggregate them under multiple norms (L1, L2, L∞) and weighting schemesthat reflect different musical traditions. I compare 12-TET with two historicallysignificant alternatives (19-TET and 31-TET) and extend the analysis to non-equaltemperaments, including quarter-comma meantone and circulating well tempera-ments. The results confirm that 12-TET is not universally optimal; its rankingdepends entirely on how the objective is defined. I then demonstrate a protocolfor inducing weights directly from the harmonic content of a composition (appliedhere to Bach’s Well-Tempered Clavier Book I, Prelude in C major), and introducea harmonic similarity metric that quantifies structural resemblance between pieces.A complete Python implementation, capable of analysing arbitrary MIDI files, isprovided.